(a) Peter and Bill are two actuaries who use the same mortality table to price fully discrete 2-year endowment insurance of 200 on (x).

- Peter calculates level annual benefit premiums of 97.4.
- Bill calculates non-level benefit premiums of 100 for the first year, and $\pi$ for the second year.
- The interest rate $i$ is 8%.
- Both actuaries calculate benefit premiums based on the equivalence principle.

The question pertains to actuarial science. Peter calculates a level premium of 97.4 for a 2-year endowment insurance, while Bill calculates non-level premiums of 100 for the first year and π for the next. Both calculations are based on the equivalence principle under an 8% interest rate.

Explanation:

This question is a case study in actuarial science, a branch of mathematics that deals with financial risks, especially in the context of insurance. In this scenario, both Peter and Bill are trying to calculate the premium for a 2-year endowment insurance policy. Peter uses level annual benefit premiums, and his calculation gives a premium of 97.4. This means the insured party pays the same premium amount every year for the lifetime of the policy.

On the other hand, Bill uses non-level benefit premiums, and his calculation gives a premium of 100 for the first year and π for the second year. With non-level premiums, the premium amount changes over the years (in this case, it decreases from year 1 to year 2).

The equivalence principle here refers to the principle that the present value of premiums should be equal to the present value of benefits, considering a certain interest rate, which in this case is 8%. Therefore, while the two actuaries are using different methods, they are both adhering to this basic principle of insurance premium calculation.

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